Precalculus Review PDF

Precalculus Review Ethan D. Bloch Revised draft August 13, 2020 + Do not circulate or post DO NOT CIRCULATE 2 DO NOT CIRCULATE Contents 1.1 Algebra...................................

Precalculus Review Ethan D. Bloch Revised draft August 13, 2020 + Do not circulate or post DO NOT CIRCULATE 2 DO NOT CIRCULATE Contents 1.1 Algebra................................................. 4 1.2 Functions and Graphs......................................... 10 1.3 Linear Functions............................................ 17 1.4 Polynomials.............................................. 20 1.5 Power Functions............................................ 23 1.6 Trigonometric Functions....................................... 26 1.7 Exponential Functions........................................ 33 1.8 Logarithmic Functions........................................ 36 3 DO NOT CIRCULATE 4 CONTENTS 1.1 Algebra Calculus makes use of precalculus—hence the name of the latter—but to do precalculus, a solid knowledge of basic algebra is needed. We review here a few of the most important ideas from algebra that are needed for calculus. Types of Numbers Precalculus, and calculus, takes place within the context of the real numbers. Within the real numbers, there are some import special types of numbers that are frequently used in mathematics. Types of Numbers 1. The real numbers, denoted R, are all the numbers on the number line, including positive num- √ bers, negative numbers, zero, whole numbers, fractions, and all other numbers (such as 2 and π). 2. The rational numbers, denoted Q, are all numbers that are expressible as fractions, for example 2 3 or −0.5. 3. The integers, denoted Z, are the numbers −4, −3, −2, −1, 0, 1, 2, 3, 4,.... 4. The natural numbers, also called the positive integers, denoted N, are the numbers 1, 2, 3, 4,.... Note that all natural numbers are integers, and all integers are rational numbers, and all rational num- bers are real numbers, but not the other way around. A collection of numbers that is even larger than the set of real numbers is the set of complex numbers, denoted C. It is not assumed that the reader is familiar with the complex numbers. These numbers are not used in Calculus I and Calculus II; they do arise in Introduction to Linear Algebra and Ordinary Differential Equations, and they will be discussed there. Infinity We will, at times, be using the symbols ∞ and −∞ to denote “infinity” and “negative infinity,” respectively. These words are written in quotes to emphasize the following. + Error Warning The symbols ∞ and −∞ are not numbers. These symbols represent what hap- pens as we take numbers that get larger and larger without bound (going to ∞) and get smaller and smaller (meaning negative numbers having larger and larger magnitude). For example, the numbers 2, 4, 8, 16, 32,... are “going to ∞,” and the numbers −1, −3, −5, −7, −9,... are “going to −∞.” DO NOT CIRCULATE 1.1. ALGEBRA 5 + Error Warning Do not try to use the symbols ∞ and −∞ in algebraic expressions (for example “∞ + 5”). Intervals Intervals are a very useful type of collections of real numbers. An interval is the set of all numbers between two fixed numbers, where the endpoints might or might not be included in the interval. The different types of interval are as follows. Intervals Let a and b be real numbers. Suppose that a ≤ b. Notation Type of Interval Definition (a, b) open bounded interval a –1, then the value of f(x) is x2. Since –2  –1, we have f(–2) = 1 – (–2) = 3. Since –1  –1, we have f(–1) = 1 – (–1) = 2. Since 2 > –1, we have f(0) = 02 = 0. 27 Example 7 – Solution cont’d How do we draw the graph of f ? We observe that if x  1, then f(x) = 1 – x, so the part of the graph of f that lies to the left of the vertical line x = –1 must coincide with the line y = 1 – x, which has slope –1 and y-intercept 1. If x > 1, then f(x) = x2, so the part of the graph of f that lies to the right of the line x = –1 must coincide with the graph of y = x2, which is a parabola. 28 Example 7 – Solution cont’d This enables us to sketch the graph in Figure 15. Figure 15 The solid dot indicates that the point (–1, 2) is included on the graph; the open dot indicates that the point (–1, 1) is excluded from the graph. 29 Piecewise Defined Functions The next example of a piecewise defined function is the absolute value function. Recall that the absolute value of a number a, denoted by |a|, is the distance from a to 0 on the real number line. Distances are always positive or 0, so we have |a|  0 for every number a For example, |3| = 3 |–3| = 3 |0| = 0 | –1| = –1 |3 –  | =  – 3 30 Piecewise Defined Functions In general, we have (Remember that if a is negative, then –a is positive.) 31 Example 8 Sketch the graph of the absolute value function f(x) = |x|. Solution: From the preceding discussion we know that x if x  0 |x| = –x if x < 0 32 Example 8 Using the same method as in Example 7, we see that the graph of f coincides with the line y = x to the right of the y-axis and coincides with the line y = –x to the left of the y-axis (see Figure 16). Figure 16 33 Example 10 We have considered the cost C(w) of mailing a large envelope with weight w. This is a piecewise defined function and we have 0.83 if 0 < w  1 1.00 if 1 < w  2 C(w) = 1.17 if 2 < w  3 1.34 if 3 < w  4 34 Example 10 cont’d The graph is shown in Figure 18. Figure 18 You can see why functions similar to this one are called step functions—they jump from one value to the next. 35 Symmetry 36 Symmetry If a function f satisfies f(–x) = f(x) for every number x in its domain, then f is called an even function. For instance, the function f(x) = x2 is even because f(–x) = (–x)2 = x2 = f(x) The geometric significance of an even function is that its graph is symmetric with respect to the y-axis (see Figure 19). An even function Figure 19 37 Symmetry This means that if we have plotted the graph of f for x  0, we obtain the entire graph simply by reflecting this portion about the y-axis. If f satisfies f(–x) = –f(x) for every number x in its domain, then f is called an odd function. For example, the function f(x) = x3 is odd because f(–x) = (–x)3 = –x3 = –f(x) 38 Symmetry The graph of an odd function is symmetric about the origin (see Figure 20). An odd function Figure 20 If we already have the graph of f for x  0, we can obtain the entire graph by rotating this portion through 180 about the origin. 39 Example 11 Determine whether each of the following functions is even, odd, or neither even nor odd. (a) f(x) = x5 + x (b) g(x) = 1 – x4 (c) h(x) = 2x – x2 Solution: (a) f(–x) = (–x)5 + (–x) = (–1)5 x5 + (–x) = –x5 – x = –(x5 + x) = –f(x) Therefore f is an odd function. 40 Example 11 – Solution cont’d (b) g(–x) = 1 – (–x4) = 1 – x4 = g(x) So g is even. (c) h(–x) = 2(–x) – (–x2) = –2x – x2 Since h(–x)  h(x) and h(–x)  –h(x), we conclude that h is neither even nor odd. 41 Symmetry The graphs of the functions in Example 11 are shown in Figure 21. Notice that the graph of h is symmetric neither about the y-axis nor about the origin. (a) (b) (c) Figure 21 42 Increasing and Decreasing Functions 43 Increasing and Decreasing Functions The graph shown in Figure 22 rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing on the interval [a, b], decreasing on [b, c], and increasing again on [c, d]. Figure 22 44 Increasing and Decreasing Functions Notice that if x1 and x2 are any two numbers between a and b with x1 < x2, then f(x1) < f(x2). We use this as the defining property of an increasing function. 45 Increasing and Decreasing Functions In the definition of an increasing function it is important to realize that the inequality f(x1) < f(x2) must be satisfied for every pair of numbers x1 and x2 in I with x1 < x2. You can see from Figure 23 that the function f(x) = x2 is decreasing on the interval (– , 0] and increasing on the interval [0, ). Figure 23 46 1 Functions and Limits Copyright © Cengage Learning. All rights reserved.  Mathematical Models: 1.2 A Catalog of Essential Functions Copyright © Cengage Learning. All rights reserved. Mathematical Models: A Catalog of Essential Functions A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. 3 Mathematical Models: A Catalog of Essential Functions Figure 1 illustrates the process of mathematical modeling. Figure 1 The modeling process 4 Mathematical Models: A Catalog of Essential Functions A mathematical model is never a completely accurate representation of a physical situation—it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say. There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs of these functions and give examples of situations appropriately modeled by such functions. 5 Linear Models 6 Linear Models When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for the function as y = f(x) = mx + b where m is the slope of the line and b is the y-intercept. 7 Linear Models A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear function f(x) = 3x – 2 and a table of sample values. Figure 2 8 Linear Models Notice that whenever x increases by 0.1, the value of f(x) increases by 0.3. So f(x) increases three times as fast as x. Thus the slope of the graph y = 3x – 2, namely 3, can be interpreted as the rate of change of y with respect to x. 9 Example 1 (a) As dry air moves upward, it expands and cools. If the ground temperature is 20C and the temperature at a height of 1 km is 10C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? 10 Example 1(a) – Solution Because we are assuming that T is a linear function of h, we can write T = mh + b We are given that T = 20 when h = 0, so 20 = m 0 + b = b In other words, the y-intercept is b = 20. We are also given that T = 10 when h = 1, so 10 = m 1 + 20 The slope of the line is therefore m = 10 – 20 = –10 and the required linear function is T = –10h + 20 11 Example 1(b) – Solution cont’d The graph is sketched in Figure 3. The slope is m = –10C/km, and this represents the rate of change of temperature with respect to height. Figure 3 12 Example 1(c) – Solution cont’d At a height of h = 2.5 km, the temperature is T = –10(2.5) + 20 = –5C 13 Linear Models If there is no physical law or principle to help us formulate a model, we construct an empirical model, which is based entirely on collected data. We seek a curve that “fits” the data in the sense that it captures the basic trend of the data points. 14 Polynomials 15 Polynomials A function P is called a polynomial if P(x) = anxn + an–1xn–1 +... + a2x2 + a1x + a0 where n is a nonnegative integer and the numbers a0, a1, a2,..., an are constants called the coefficients of the polynomial. The domain of any polynomial is If the leading coefficient an  0, then the degree of the polynomial is n. For example, the function is a polynomial of degree 6. 16 Polynomials A polynomial of degree 1 is of the form P(x) = mx + b and so it is a linear function. A polynomial of degree 2 is of the form P(x) = ax2 + bx + c and is called a quadratic function. 17 Polynomials Its graph is always a parabola obtained by shifting the parabola y = ax2. The parabola opens upward if a > 0 and downward if a < 0. (See Figure 7.) The graphs of quadratic functions are parabolas. Figure 7 18 Polynomials A polynomial of degree 3 is of the form P(x) = ax3 + bx2 + cx + d a0 and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). Figure 8 19 Example 4 A ball is dropped from the upper observation deck of the CN Tower, 450m above the ground, and its height h above the ground is recorded at 1-second intervals in Table 2. Find a model to fit the data and use the model to predict the time at which the ball hits the ground. 20 Example 4 – Solution We draw a scatter plot of the data in Figure 9 and observe that a linear model is inappropriate. Scatter plot for a falling ball Figure 9 21 Example 4 – Solution cont’d But it looks as if the data points might lie on a parabola, so we try a quadratic model instead. Using a graphing calculator or computer algebra system (which uses the least squares method), we obtain the following quadratic model: h = 449.36 + 0.96ts – 4.90t2 22 Example 4 – Solution cont’d In Figure 10 we plot the graph of Equation 3 together with the data points and see that the quadratic model gives a very good fit. Quadratic model for a falling ball Figure 10 The ball hits the ground when h = 0, so we solve the quadratic equation –4.90t2 + 0.96t + 449.36 = 0 23 Example 4 – Solution cont’d The quadratic formula gives The positive root is t  9.67, so we predict that the ball will hit the ground after about 9.7 seconds. 24 Power Functions 25 Power Functions A function of the form f(x) = xa, where a is a constant, is called a power function. We consider several cases. (i) a = n, where n is a positive integer The graphs of f(x) = xn for n = 1, 2, 3, 4, and 5 are shown in Figure 11. (These are polynomials with only one term.) We already know the shape of the graphs of y = x (a line through the origin with slope 1) and y = x2 (a parabola). 26 Power Functions Graphs of f(x) = xn for n = 1, 2, 3, 4, 5 Figure 11 27 Power Functions The general shape of the graph of f(x) = xn depends on whether n is even or odd. If n is even, then f(x) = xn is an even function and its graph is similar to the parabola y = x2. If n is odd, then f(x) = xn is an odd function and its graph is similar to that of y = x3. 28 Power Functions Notice from Figure 12, however, that as n increases, the graph of y = xn becomes flatter near 0 and steeper when |x|  1. (If x is small, then x2 is smaller, x3 is even smaller, x4 is smaller still, and so on.) Families of power functions Figure 12 29 Power Functions (ii) a = 1/n, where n is a positive integer The function is a root function. For n = 2 it is the square root function whose domain is [0, ) and whose graph is the upper half of the parabola x = y2. [See Figure 13(a).] Graph of root function Figure 13(a) 30 Power Functions For other even values of n, the graph of is similar to that of For n = 3 we have the cube root function whose domain is (recall that every real number has a cube root) and whose graph is shown in Figure 13(b). The graph of for n odd (n > 3) is similar to that of Graph of root function Figure 13(b) 31 Power Functions (iii) a = –1 The graph of the reciprocal function f(x) = x –1 = 1/x is shown in Figure 14. Its graph has the equation y = 1/x, or xy = 1, and is a hyperbola with the coordinate axes as its asymptotes. The reciprocal function Figure 14 32 Power Functions This function arises in physics and chemistry in connection with Boyle’s Law, which says that, when the temperature is constant, the volume V of a gas is inversely proportional to the pressure P: where C is a constant. Thus the graph of V as a function of P (see Figure 15) has the same general shape as the right half of Figure 14. Volume as a function of pressure at constant temperature Figure 15 33 Rational Functions 34 Rational Functions A rational function f is a ratio of two polynomials: where P and Q are polynomials. The domain consists of all values of x such that Q(x)  0. A simple example of a rational function is the function f(x) = 1/x, whose domain is {x|x  0}; this is the reciprocal function graphed in Figure 14. The reciprocal function Figure 14 35 Rational Functions The function is a rational function with domain {x|x  2}. Its graph is shown in Figure 16. Figure 16 36 Algebraic Functions 37 Algebraic Functions A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. Here are two more examples: 38 Algebraic Functions The graphs of algebraic functions can assume a variety of shapes. Figure 17 illustrates some of the possibilities. Figure 17 39 Algebraic Functions An example of an algebraic function occurs in the theory of relativity. The mass of a particle with velocity v is where m0 is the rest mass of the particle and c = 3.0 x 105 km/s is the speed of light in a vacuum. 40 Trigonometric Functions 41 Trigonometric Functions In calculus the convention is that radian measure is always used (except when otherwise indicated). For example, when we use the function f(x) = sin x, it is understood that sin x means the sine of the angle whose radian measure is x. 42 Trigonometric Functions Thus the graphs of the sine and cosine functions are as shown in Figure 18. Figure 18 43 Trigonometric Functions Notice that for both the sine and cosine functions the domain is ( , ) and the range is the closed interval [–1, 1]. Thus, for all values of x, we have or, in terms of absolute values, |sin x|  1 |cos x|  1 44 Trigonometric Functions Also, the zeros of the sine function occur at the integer multiples of  ; that is, sin x = 0 when x = n n an integer An important property of the sine and cosine functions is that they are periodic functions and have period 2. This means that, for all values of x, 45 Trigonometric Functions The tangent function is related to the sine and cosine functions by the equation and its graph is shown in Figure 19. It is undefined whenever cos x = 0, that is, when x =  /2, 3 /2,.... y = tan x Figure 19 Its range is ( , ). 46 Trigonometric Functions Notice that the tangent function has period  : tan(x + ) = tan x for all x The remaining three trigonometric functions (cosecant, secant, and cotangent) are the reciprocals of the sine, cosine, and tangent functions. 47 Exponential Functions 48 Exponential Functions The exponential functions are the functions of the form f(x) = ax, where the base a is a positive constant. The graphs of y = 2x and y = (0.5)x are shown in Figure 20. In both cases the domain is ( , ) and the range is (0, ). Figure 20 49 Exponential Functions Exponential functions are useful for modeling many natural phenomena, such as population growth (if a > 1) and radioactive decay (if a < 1). 50 Logarithmic Functions 51 Logarithmic Functions The logarithmic functions f(x) = logax, where the base a is a positive constant, are the inverse functions of the exponential functions. Figure 21 shows the graphs of four logarithmic functions with various bases. In each case the domain is (0, ), the range is ( , ), and the function increases slowly when x > 1. Figure 21 52 Example 5 Classify the following functions as one of the types of functions that we have discussed. (a) f(x) = 5x (b) g(x) = x5 (c) (d) u(t) = 1 – t + 5t 4 53 Example 5 – Solution (a) f(x) = 5x is an exponential function. (The x is the exponent.) (b) g(x) = x5 is a power function. (The x is the base.) We could also consider it to be a polynomial of degree 5. (c) is an algebraic function. (d) u(t) = 1 – t + 5t 4 is a polynomial of degree 4. 54 1 Functions and Limits Copyright © Cengage Learning. All rights reserved. 1 Combinations of Functions 14 Combinations of Functions Two functions f and g can be combined to form new functions f + g, f – g, fg, and f/g in a manner similar to the way we add, subtract, multiply, and divide real numbers. The sum and difference functions are defined by (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x) If the domain of f is A and the domain of g is B, then the domain of f + g is the intersection A ∩ B because both f(x) and g(x) have to be defined. For example, the domain of is A = [0, ) and the domain of is B = ( , 2], so the domain of is A ∩ B = [0, 2]. 15 Combinations of Functions Similarly, the product and quotient functions are defined by The domain of fg is A ∩ B, but we can’t divide by 0 and so the domain of f/g is {x  A ∩ B | g(x)  0}. For instance, if f(x) = x2 and g(x) = x – 1, then the domain of the rational function (f/g)(x) = x2/(x – 1) is {x | x  1}, or ( , 1) U (1, ). 16 Combinations of Functions There is another way of combining two functions to obtain a new function. For example, suppose that y = f(u) = and u = g(x) = x2 + 1. Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x. We compute this by substitution: y = f(u) = f(g(x)) = f(x2 + 1) = The procedure is called composition because the new function is composed of the two given functions f and g. 17 Combinations of Functions In general, given any two functions f and g, we start with a number x in the domain of g and find its image g(x). If this number g(x) is in the domain of f, then we can calculate the value of f(g(x)). The result is a new function h(x) = f(g(x)) obtained by substituting g into f. It is called the composition (or composite) of f and g and is denoted by f  g (“f circle g”). 18 Combinations of Functions The domain of f  g is the set of all x in the domain of g such that g(x) is in the domain of f. In other words, (f  g)(x) is defined whenever both g(x) and f(g(x)) are defined. Figure 11 shows how to picture f  g in terms of machines. The f  g machine is composed of the g machine (first) and then the f machine. Figure 11 19 Example 6 If f(x) = x2 and g(x) = x – 3, find the composite functions f  g and g  f. Solution: We have (f  g)(x) = f(g(x)) = f(x – 3) = (x – 3)2 (g  f)(x) = g(f(x)) = g(x2) = x2 – 3 20 Combinations of Functions Remember, the notation f  g means that the function g is applied first and then f is applied second. In Example 6, f  g is the function that first subtracts 3 and then squares; g  f is the function that first squares and then subtracts 3. It is possible to take the composition of three or more functions. For instance, the composite function f  g  h is found by first applying h, then g, and then f as follows: (f  g  h)(x) = f(g(h(x))) 21 1 Functions and Limits Copyright © Cengage Learning. All rights reserved. 1  New Functions from Old 1.3 Functions Copyright © Cengage Learning. All rights reserved. Transformations of Functions By applying certain transformations to the graph of a given function we can obtain the graphs of certain related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs. Let’s first consider translations. If c is a positive number, then the graph of y = f(x) + c is just the graph of y = f(x) shifted upward a distance of c units (because each y-coordinate is increased by the same number c). 4 Transformations of Functions Likewise, if g(x) = f(x – c), where c > 0, then the value of g at x is the same as the value of f at x – c (c units to the left of x). Therefore the graph of y = f(x – c), is just the graph of y = f(x) shifted c units to the right (see Figure 1). Translating the graph of ƒ Figure 1 5 Transformations of Functions Now let’s consider the stretching and reflecting transformations. If c > 1, then the graph of y = cf(x) is the graph of y = f(x) stretched by a factor of c in the vertical direction (because each y-coordinate is multiplied by the same number c). 6 Transformations of Functions The graph of y = –f(x) is the graph of y = f(x) reflected about the x-axis because the point (x, y) is replaced by the point (x, –y). (See Figure 2 and the following chart, where the results of other stretching, shrinking, and reflecting transformations are also given.) Stretching and reflecting the graph of f Figure 2 7 Transformations of Functions 8 Transformations of Functions Figure 3 illustrates these stretching transformations when applied to the cosine function with c = 2. Figure 3 9 Transformations of Functions For instance, in order to get the graph of y = 2 cos x we multiply the y-coordinate of each point on the graph of y = cos x by 2. This means that the graph of y = cos x gets stretched vertically by a factor of 2. 10 Example 1 Given the graph of use transformations to graph and Solution: The graph of the square root function , is shown in Figure 4(a). Figure 4 11 Example 1 – Solution cont’d Figure 4 In the other parts of the figure we sketch by shifting 2 units downward, by shifting 2 units to the right, by reflecting about the x-axis, by stretching vertically by a factor of 2, and by reflecting about the y-axis. 12 Transformations of Functions Another transformation of some interest is taking the absolute value of a function. If y = |f(x)|, then according to the definition of absolute value, y = f(x) when f(x) ≥ 0 and y = –f(x) when f(x) < 0. This tells us how to get the graph of y = |f(x)| from the graph of y = f(x): The part of the graph that lies above the x-axis remains the same; the part that lies below the x-axis is reflected about the x-axis. 13 1 Functions and Limits Copyright © Cengage Learning. All rights reserved. 1.5 The Limit of a Function Copyright © Cengage Learning. All rights reserved. The Limit of a Function To find the tangent to a curve or the velocity of an object, we now turn our attention to limits in general and numerical and graphical methods for computing them. Let’s investigate the behavior of the function f defined by f(x) = x2 – x + 2 for values of x near 2. 3 The Limit of a Function The following table gives values of f(x) for values of x close to 2 but not equal to 2. 4 The Limit of a Function From the table and the graph of f (a parabola) shown in Figure 1 we see that when x is close to 2 (on either side of 2), f(x) is close to 4. Figure 1 5 The Limit of a Function In fact, it appears that we can make the values of f(x) as close as we like to 4 by taking x sufficiently close to 2. We express this by saying “the limit of the function f(x) = x2 – x + 2 as x approaches 2 is equal to 4.” The notation for this is 6 The Limit of a Function In general, we use the following notation. This says that the values of f(x) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a) but x  a. 7 The Limit of a Function An alternative notation for is f(x) L as xa which is usually read “f(x) approaches L as x approaches a.” Notice the phrase “but x  a” in the definition of limit. This means that in finding the limit of f(x) as x approaches a, we never consider x = a. In fact, f(x) need not even be defined when x = a. The only thing that matters is how f is defined near a. 8 The Limit of a Function Figure 2 shows the graphs of three functions. Note that in part (c), f(a) is not defined and in part (b), f(a)  L. But in each case, regardless of what happens at a, it is true that limxa f(x) = L. in all three cases Figure 2 9 Example 1 Guess the value of Solution: Notice that the function f(x) = (x – 1)(x2 – 1) is not defined when x = 1, but that doesn’t matter because the definition of limxa f(x) says that we consider values of x that are close to a but not equal to a. 10 Example 1 – Solution cont’d The tables below give values of f(x) (correct to six decimal places) for values of x that approach 1 (but are not equal to 1). On the basis of the values in the tables, we make the guess that 11 The Limit of a Function Example 1 is illustrated by the graph of f in Figure 3. Now let’s change f slightly by giving it the value 2 when x = 1 and calling the resulting function g: Figure 3 12 The Limit of a Function This new function g still has the same limit as x approaches 1. (See Figure 4.) Figure 4 13 One-Sided Limits 14 One-Sided Limits The function H is defined by. H(t) approaches 0 as t approaches 0 from the left and H(t) approaches 1 as t approaches 0 from the right. We indicate this situation symbolically by writing and 15 One-Sided Limits The symbol “t 0–” indicates that we consider only values of t that are less than 0. Likewise, “t 0+” indicates that we consider only values of t that are greater than 0. 16 One-Sided Limits Notice that Definition 2 differs from Definition 1 only in that we require x to be less than a. 17 One-Sided Limits Similarly, if we require that x be greater than a, we get “the right-hand limit of f(x) as x approaches a is equal to L” and we write Thus the symbol “x a+” means that we consider only x > a. These definitions are illustrated in Figure 9. Figure 9 18 One-Sided Limits By comparing Definition 1 with the definitions of one-sided limits, we see that the following is true. 19 Example 7 The graph of a function g is shown in Figure 10. Use it to state the values (if they exist) of the following: Figure 10 20 Example 7 – Solution From the graph we see that the values of g(x) approach 3 as x approaches 2 from the left, but they approach 1 as x approaches 2 from the right. Therefore and (c) Since the left and right limits are different, we conclude from that limx2 g(x) does not exist. 21 Example 7 – Solution cont’d The graph also shows that and (f) This time the left and right limits are the same and so, by , we have Despite this fact, notice that g(5)  2. 22 Infinite Limits 23 Infinite Limits Another notation for limxa f(x) =  is f(x)  as x a 24 Infinite Limits Again, the symbol  is not a number, but the expression limxa f(x) =  is often read as “the limit of f(x), as approaches a, is infinity” or “f(x) becomes infinite as approaches a” or “f(x) increases without bound as approaches a” 25 Infinite Limits This definition is illustrated graphically in Figure 12. Figure 12 26 Infinite Limits A similar sort of limit, for functions that become large negative as x gets close to a, is defined in Definition 5 and is illustrated in Figure 13. Figure 13 27 Infinite Limits The symbol limxa f(x) = – can be read as “the limit of f (x), as x approaches a, is negative infinity” or “f(x) decreases without bound as x approaches a.” As an example we have 28 Infinite Limits Similar definitions can be given for the one-sided infinite limits remembering that “x a–” means that we consider only values of that are less than a, and similarly “x a+” means that we consider only x > a. 29 Infinite Limits Illustrations of these four cases are given in Figure 14. Figure 14 30 Infinite Limits 31 Example 10 Find the vertical asymptotes of f(x) = tan x. Solution: Because there are potential vertical asymptotes where cos x = 0. In fact, since cos x 0+ as x ( /2)+, whereas sin x is positive when x is near  /2, we have 32 Example 10 – Solution cont’d This shows that the line x =  /2 is a vertical asymptote. Similar reasoning shows that the lines x = (2n + 1) /2, where n is an integer, are all vertical asymptotes of f(x) = tan x. The graph in Figure 16 confirms this. y = tan x Figure 16 33 1 Functions and Limits Copyright © Cengage Learning. All rights reserved.  Calculating Limits Using 1.6 the Limit Laws Copyright © Cengage Learning. All rights reserved. Calculating Limits Using the Limit Laws In this section we use the following properties of limits, called the Limit Laws, to calculate limits. 3 Calculating Limits Using the Limit Laws These five laws can be stated verbally as follows: Sum Law 1. The limit of a sum is the sum of the limits. Difference Law 2. The limit of a difference is the difference of the limits. Constant Multiple Law 3. The limit of a constant times a function is the constant times the limit of the function. 4 Calculating Limits Using the Limit Laws Product Law 4. The limit of a product is the product of the limits. Quotient Law 5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0). For instance, if f(x) is close to L and g(x) is close to M, it is reasonable to conclude that f(x) + g(x) is close to L + M. 5 Example 1 Use the Limit Laws and the graphs of f and g in Figure 1 to evaluate the following limits, if they exist. Figure 1 6 Example 1(a) – Solution From the graphs of f and g we see that and Therefore we have (by Law 1) (by Law 3) = 1 + 5(–1) = –4 7 Example 1(b) – Solution cont’d We see that limx 1 f(x) = 2. But limx 1 g(x) does not exist because the left and right limits are different: So we can’t use Law 4 for the desired limit. But we can use Law 4 for the one-sided limits: The left and right limits aren’t equal, so limx 1 [f(x)g(x)] does not exist. 8 Example 1(c) – Solution cont’d The graphs show that and Because the limit of the denominator is 0, we can’t use Law 5. Figure 1 The given limit does not exist because the denominator approaches 0 while the numerator approaches a nonzero number. 9 Calculating Limits Using the Limit Laws If we use the Product Law repeatedly with g(x) = f(x), we obtain the following law. Power Law In applying these six limit laws, we need to use two special limits: These limits are obvious from an intuitive point of view (state them in words or draw graphs of y = c and y = x). 10 Calculating Limits Using the Limit Laws If we now put f(x) = x in Law 6 and use Law 8, we get another useful special limit. A similar limit holds for roots as follows. More generally, we have the following law. Root Law 11 Calculating Limits Using the Limit Laws Functions with the Direct Substitution Property are called continuous at a. In general, we have the following useful fact. 12 Calculating Limits Using the Limit Laws Some limits are best calculated by first finding the left- and right-hand limits. The following theorem says that a two-sided limit exists if and only if both of the one-sided limits exist and are equal. When computing one-sided limits, we use the fact that the Limit Laws also hold for one-sided limits. 13 Example 10 The greatest integer function is defined by the largest integer that is less than or equal to x. (For instance, ) Show that does not exist. Solution: The graph of the greatest integer function is shown in Figure 6. Since for 3  x < 4, we have Greatest integer function Figure 6 14 Example 10 – Solution cont’d Since for 2  x < 3, we have Because these one-sided limits are not equal, does not exist by Theorem 1. 15 Calculating Limits Using the Limit Laws The next two theorems give two additional properties of limits. 16 Calculating Limits Using the Limit Laws The Squeeze Theorem, which is sometimes called the Sandwich Theorem or the Pinching Theorem, is illustrated by Figure 7. It says that if g(x) is squeezed between f(x) and h(x) near a, and if f and h have the same limit L at a, then g is forced to have the same limit L at a. Figure 7 17 1 Functions and Limits Copyright © Cengage Learning. All rights reserved. 1.8 Continuity Copyright © Cengage Learning. All rights reserved. Continuity The limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Functions with this property are called continuous at a. We will see that the mathematical definition of continuity corresponds closely with the meaning of the word continuity in everyday language. (A continuous process is one that takes place gradually, without interruption or abrupt change.) 3 Continuity Notice that Definition 1 implicitly requires three things if f is continuous at a: 1. f(a) is defined (that is, a is in the domain of f ) 2. exists 3. The definition says that f is continuous at a if f(x) approaches f(a) as x approaches a. Thus a continuous function f has the property that a small change in x produces only a small change in f(x). 4 Continuity In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not continuous at a. Physical phenomena are usually continuous. For instance, the displacement or velocity of a vehicle varies continuously with time, as does a person’s height. But discontinuities do occur in such situations as electric currents. 5 Continuity Geometrically, you can think of a function that is continuous at every number in an interval as a function whose graph has no break in it. The graph can be drawn without removing your pen from the paper. 6 Example 1 Figure 2 shows the graph of a function f. At which numbers is f discontinuous? Why? Figure 2 Solution: It looks as if there is a discontinuity when a = 1 because the graph has a break there. The official reason that f is discontinuous at 1 is that f(1) is not defined. 7 Example 1 – Solution cont’d The graph also has a break when a = 3, but the reason for the discontinuity is different. Here, f(3) is defined, but limx3 f(x) does not exist (because the left and right limits are different). So f is discontinuous at 3. What about a = 5? Here, f(5) is defined and limx5 f(x) exists (because the left and right limits are the same). But So f is discontinuous at 5. 8 Example 2 Where are each of the following functions discontinuous? Solution: (a) Notice that f(2) is not defined, so f is discontinuous at 2. Later we’ll see why f is continuous at all other numbers. 9 Example 2 – Solution cont’d (b) Here f(0) = 1 is defined but does not exist. So f is discontinuous at 0. (c) Here f(2) = 1 is defined and = 3 exists. 10 Example 2 – Solution cont’d But so f is not continuous at 2. (d) The greatest integer function f(x) = has discontinuities at all of the integers because does not exist if n is an integer. 11 Continuity Figure 3 shows the graphs of the functions in Example 2. Graphs of the functions in Example 2 Figure 3 12 Continuity Graphs of the functions in Example 2 Figure 3 13 Continuity In each case the graph can’t be drawn without lifting the pen from the paper because a hole or break or jump occurs in the graph. The kind of discontinuity illustrated in parts (a) and (c) is called removable because we could remove the discontinuity by redefining f at just the single number 2. [The function g(x) = x + 1 is continuous.] The discontinuity in part (b) is called an infinite discontinuity. The discontinuities in part (d) are called jump discontinuities because the function “jumps” from one value to another. 14 Continuity 15 Continuity Instead of always using Definitions 1, 2, and 3 to verify the continuity of a function, it is often convenient to use the next theorem, which shows how to build up complicated continuous functions from simple ones. 16 Continuity It follows from Theorem 4 and Definition 3 that if f and g are continuous on an interval, then so are the functions f + g, f – g, cf, fg, and (if g is never 0) f/g. The following theorem was stated as the Direct Substitution Property. 17 Continuity As an illustration of Theorem 5, observe that the volume of a sphere varies continuously with its radius because the formula V(r) =  r 3 shows that V is a polynomial function of r. Likewise, if a ball is thrown vertically into the air with a velocity of 50 ft/s, then the height of the ball in feet t seconds later is given by the formula h = 50t – 16t2. Again this is a polynomial function, so the height is a continuous function of the elapsed time. 18 Continuity It turns out that most of the familiar functions are continuous at every number in their domains. From the appearance of the graphs of the sine and cosine functions, we would certainly guess that they are continuous. We know from the definitions of sin  and cos  that the coordinates of the point P in Figure 5 are (cos , sin  ). As  0, we see that P approaches the point (1, 0) and so cos  1 and sin  0. Figure 5 19 Continuity Thus Since cos 0 = 1 and sin 0 = 0, the equations in assert that the cosine and sine functions are continuous at 0. The addition formulas for cosine and sine can then be used to deduce that these functions are continuous everywhere. It follows from part 5 of Theorem 4 that is continuous except where cos x = 0. 20 Continuity This happens when x is an odd integer multiple of  /2, so y = tan x has infinite discontinuities when x =  /2, 3 /2, 5 /2, and so on (see Figure 6). y = tan x Figure 6 21 Continuity Another way of combining continuous functions f and g to get a new continuous function is to form the composite function f  g. This fact is a consequence of the following theorem. 22 Continuity Intuitively, Theorem 8 is reasonable because if x is close to a, then g(x) is close to b, and since f is continuous at b, if g(x) is close to b, then f(g(x)) is close to f(b). An important property of continuous functions is expressed by the following theorem, whose proof is found in more advanced books on calculus. 23 Continuity The Intermediate Value Theorem states that a continuous function takes on every intermediate value between the function values f(a) and f(b). It is illustrated by Figure 7. Note that the value N can be taken on once [as in part (a)] or more than once [as in part (b)]. Figure 7 24 Continuity If we think of a continuous function as a function whose graph has no hole or break, then it is easy to believe that the Intermediate Value Theorem is true. In geometric terms it says that if any horizontal line y = N is given between y = f(a) and y = f(b) as in Figure 8, then the graph of f can’t jump over the line. It must intersect y = N somewhere. Figure 8 25 Continuity It is important that the function f in Theorem 10 be continuous. The Intermediate Value Theorem is not true in general for discontinuous functions. We can use a graphing calculator or computer to illustrate the use of the Intermediate Value Theorem. Figure 9 shows the graph of f in the viewing rectangle [–1, 3] by [–3, 3] and you can see that the graph crosses the x-axis between 1 and 2. Figure 9 26 Continuity Figure 10 shows the result of zooming in to the viewing rectangle [1.2, 1.3] by [–0.2, 0.2]. Figure 10 In fact, the Intermediate Value Theorem plays a role in the very way these graphing devices work. 27 Continuity A computer calculates a finite number of points on the graph and turns on the pixels that contain these calculated points. It assumes that the function is continuous and takes on all the intermediate values between two consecutive points. The computer therefore connects the pixels by turning on the intermediate pixels. 28 2 Derivatives Copyright © Cengage Learning. All rights reserved. 2.3 Differentiation Formulas Copyright © Cengage Learning. All rights reserved. Differentiation Formulas Let’s start with the simplest of all functions, the constant function f(x) = c. The graph of this function is the horizontal line y = c, which has slope 0, so we must have f(x) = 0. See Figure 1. The graph of f(x) = c is the line y = c, so f(x) = 0. Figure 1 3 Differentiation Formulas A formal proof, from the definition of a derivative, is also easy: In Leibniz notation, we write this rule as follows. 4 Power Functions 5 Power Functions We next look at the functions f(x) = xn, where n is a positive integer. If n = 1, the graph of f(x) = x is the line y = x, which has slope 1. (See Figure 2.) The graph of f(x) = x is the line y = x, so f (x) = 1. Figure 2 6 Power Functions So We have already investigated the cases n = 2 and n = 3. In fact, we found that (x2) = 2x (x3) = 3x2 7 Power Functions For n = 4 we find the derivative of f(x) = x4 as follows: 8 Power Functions Thus (x4) = 4x3 Comparing the equations in , , and , we see a pattern emerging. It seems to be a reasonable guess that, when n is a positive integer, (ddx)(xn) = nx n–1. This turns out to be true. We prove it in two ways; the second proof uses the Binomial Theorem. 9 Example 1 (a) If f(x) = x6, then f(x) = 6x5. (b) If y = x1000, then y = 1000x999. (c) If y = t 4, then = 4t 3. (d) (r3) = 3r2 10 New Derivatives from Old 11 New Derivatives from Old When new functions are formed from old functions by addition, subtraction, or multiplication by a constant, their derivatives can be calculated in terms of derivatives of the old functions. In particular, the following formula says that the derivative of a constant times a function is the constant times the derivative of the function. 12 Example 2 (a) (3x4) = 3 (x4) = 3(4x3) = 12x3 (b) (–x) = [(–1)x] = (–1) (x) = –1(1) = –1 13 New Derivatives from Old The next rule tells us that the derivative of a sum of functions is the sum of the derivatives. The Sum Rule can be extended to the sum of any number of functions. For instance, using this theorem twice, we get (f + g + h) = [(f + g) + h] = (f + g) + h = f + g + h 14 New Derivatives from Old By writing f – g as f + (–1)g and applying the Sum Rule and the Constant Multiple Rule, we get the following formula. The Constant Multiple Rule, the Sum Rule, and the Difference Rule can be combined with the Power Rule to differentiate any polynomial, as the following examples demonstrate. 15 Example 3 (x8 + 12x5 – 4x4 + 10x3 – 6x + 5) = (x8) + 12 (x5) – 4 (x4) + 10 (x3) – 6 (x) + (5) = 8x7 + 12(5x4) – 4(4x3) + 10(3x2) – 6(1) + 0 = 8x7 + 60x4 – 16x3 + 30x2 – 6 16 New Derivatives from Old Next we need a formula for the derivative of a product of two functions. By analogy with the Sum and Difference Rules, one might be tempted to guess, as Leibniz did three centuries ago, that the derivative of a product is the product of the derivatives. We can see, however, that this guess is wrong by looking at a particular example. Let f(x) = x and g(x) = x2. Then the Power Rule gives f(x) = 1 and g(x) = 2x. But (fg)(x) = x3, so (fg)(x) = 3x2. Thus (fg)  fg. 17 New Derivatives from Old The correct formula was discovered by Leibniz and is called the Product Rule. In words, the Product Rule says that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. 18 Example 6 Find F(x) if F(x) = (6x3)(7x4). Solution: By the Product Rule, we have F(x) = = (6x3)(28x3) + (7x4)(18x2) = 168x6 + 126x6 = 294x6 19 New Derivatives from Old In words, the Quotient Rule says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. 20 Example 8 Let. Then 21 New Derivatives from Old Note: Don’t use the Quotient Rule every time you see a quotient. Sometimes it’s easier to rewrite a quotient first to put it in a form that is simpler for the purpose of differentiation. For instance, although it is possible to differentiate the function F(x) = using the Quotient Rule. 22 New Derivatives from Old It is much easier to perform the division first and write the function as F(x) = 3x + 2x –12 before differentiating. 23 General Power Functions 24 General Power Functions The Quotient Rule can be used to extend the Power Rule to the case where the exponent is a negative integer. 25 Example 9 (a) If y = , then = –x –2 = (b) 26 General Power Functions 27 Example 11 Differentiate the function f(t) = (a + bt). Solution 1: Using the Product Rule, we have 28 Example 11 – Solution 2 cont’d If we first use the laws of exponents to rewrite f(t), then we can proceed directly without using the Product Rule. 29 General Power Functions The differentiation rules enable us to find tangent lines without having to resort to the definition of a derivative. They also enable us to find normal lines. The normal line to a curve C at point P is the line through P that is perpendicular to the tangent line at P. 30 Example 12 Find equations of the tangent line and normal line to the curve y = (1 + x2) at the point (1, ). Solution: According to the Quotient Rule, we have 31 Example 12 – Solution cont’d So the slope of the tangent line at (1, ) is We use the point-slope form to write an equation of the tangent line at (1, ): y– = – (x – 1) or y= 32 Example 12 – Solution cont’d The slope of the normal line at (1, ) is the negative reciprocal of , namely 4, so an equation is y– = 4(x – 1) or y = 4x – The curve and its tangent and normal lines are graphed in Figure 5. Figure 5 33 General Power Functions We summarize the differentiation formulas we have learned so far as follows. Table of Differentiation Formulas 34 2 Derivatives Copyright © Cengage Learning. All rights reserved. 2.5 The Chain Rule Copyright © Cengage Learning. All rights reserved. The Chain Rule Suppose you are asked to differentiate the function The differentiation formulas you learned in the previous sections of this chapter do not enable you to calculate F(x). Observe that F is a composite function. In fact, if we let y = f (u) = and let u = g(x) = x2 + 1, then we can write y = F (x) = f (g(x)), that is, F = f  g. We know how to differentiate both f and g, so it would be useful to have a rule that tells us how to find the derivative of F = f  g in terms of the derivatives of f and g. 3 The Chain Rule It turns out that the derivative of the composite function f  g is the product of the derivatives of f and g. This fact is one of the most important of the differentiation rules and is called the Chain Rule. It seems plausible if we interpret derivatives as rates of change. Regard du/dx as the rate of change of u with respect to x, dy/du as the rate of change of y with respect to u, and dy/dx as the rate of change of y with respect to x. If u changes twice as fast as x and y changes three times as fast as u, then it seems reasonable that y changes six times as fast as x, and so we expect that 4 The Chain Rule 5 The Chain Rule The Chain Rule can be written either in the prime notation (f  g)(x) = f(g(x))  g(x) or, if y = f(u) and u = g(x), in Leibniz notation: Equation 3 is easy to remember because if dy/du and du/dx were quotients, then we could cancel du. Remember, however, that du has not been defined and du/dx should not be thought of as an actual quotient. 6 Example 1 Find F'(x) if F(x) =. Solution 1: (Using Equation 2): We have expressed F as F(x) = (f  g)(x) = f(g(x)) where f(u) = and g(x) = x2 + 1. Since and g(x) = 2x we have F(x) = f(g(x))  g(x) 7 Example 1 – Solution cont’d (Using Equation 3): If we let u = x2 + 1 and y = , then 8 The Chain Rule When using Formula 3 we should bear in mind that dy/dx refers to the derivative of y when y is considered as a function of x (called the derivative of y with respect to x), whereas dy/du refers to the derivative of y when considered as a function of u (the derivative of y with respect to u). For instance, in Example 1, y can be considered as a function of x (y = ) and also as a function of u (y = ). Note that whereas 9 The Chain Rule In general, if y = sin u, where u is a differentiable function of x, then, by the Chain Rule, Thus In a similar fashion, all of the formulas for differentiating trigonometric functions can be combined with the Chain Rule. 10 The Chain Rule Let’s make explicit the special case of the Chain Rule where the outer function f is a power function. If y = [g(x)]n, then we can write y = f(u) = un where u = g(x). By using the Chain Rule and then the Power Rule, we get 11 Example 3 Differentiate y = (x3 – 1)100. Solution: Taking u = g(x) = x3 – 1 and n = 100 in , we have = (x3 – 1)100 = 100(x3 – 1)99 (x3 – 1) = 100(x3 – 1)99  3x2 = 300x2(x3 – 1)99 12 How to Prove the Chain Rule 13 How to Prove the Chain Rule Recall that if y = f(x) and x changes from a to a + x, we defined the increment of y as y = f(a + x) – f(a) According to the definition of a derivative, we have So if we denote by ε the difference between the difference quotient and the derivative, we obtain = f'(a) – f'(a) = 0 14 How to Prove the Chain Rule But y = f(a) x + ε x If we define ε to be 0 when x = 0, then ε becomes a continuous function of x. Thus, for a differentiable function f, we can write y = f(a) x + ε x where ε 0 as x 0 and ε is a continuous function of x. This property of differentiable functions is what enables us to prove the Chain Rule. 15 2 Derivatives Copyright © Cengage Learning. All rights reserved. 2.1 Derivatives and Rates of Change Copyright © Cengage Learning. All rights reserved. Derivatives and Rates of Change The problem of finding the tangent line to a curve and the problem of finding the velocity of an object both involve finding the same type of limit. This special type of limit is called a derivative and we will see that it can be interpreted as a rate of change in any of the sciences or engineering. 3 Tangents 4 Tangents If a curve C has equation y = f(x) and we want to find the tangent line to C at the point P(a, f(a)), then we consider a nearby point Q(x, f(x)), where x  a, and compute the slope of the secant line PQ: Then we let Q approach P along the curve C by letting x approach a. 5 Tangents If mPQ approaches a number m, then we define the tangent t to be the line through P with slope m. (This amounts to saying that the tangent line is the limiting position of the secant line PQ as Q approaches P. See Figure 1.) Figure 1 6 Tangents 7 Example 1 Find an equation of the tangent line to the parabola y = x2 at the point P(1, 1). Solution: Here we have a = 1 and f(x) = x2, so the slope is 8 Example 1 – Solution cont’d Using the point-slope form of the equation of a line, we find that an equation of the tangent line at (1, 1) is y – 1 = 2(x – 1) or y = 2x – 1 9 Tangents We sometimes refer to the slope of the tangent line to a curve at a point as the slope of the curve at the point. The idea is that if we zoom in far enough toward the point, the curve looks almost like a straight line. Figure 2 illustrates this procedure for the curve y = x2 in Example 1. Zooming in toward the point (1, 1) on the parabola y = x2 Figure 2 10 Tangents The more we zoom in, the more the parabola looks like a line. In other words, the curve becomes almost indistinguishable from its tangent line. There is another expression for the slope of a tangent line that is sometimes easier to use. 11 Tangents If h = x – a, then x = a + h and so the slope of the secant line PQ is (See Figure 3 where the case h > 0 is illustrated and Q is to the right of P. If it happened that h < 0, however, Q would be to the left of P.) Figure 3 12 Tangents Notice that as x approaches a, h approaches 0 (because h = x – a) and so the expression for the slope of the tangent line in Definition 1 becomes 13 Velocities 14 Velocities In general, suppose an object moves along a straight line according to an equation of motion s = f(t), where s is the displacement (directed distance) of the object from the origin at time t. The function f that describes the motion is called the position function of the object. In the time interval from t = a to t = a + h the change in position is f(a + h) – f(a). (See Figure 5.) Figure 5 15 Velocities The average velocity over this time interval is 16 Velocities which is the same as the slope of the secant line PQ in Figure 6. Figure 6 17 Velocities Now suppose we compute the average velocities over shorter and shorter time intervals [a, a + h]. In other words, we let h approach 0. As in the example of the falling ball, we define the velocity (or instantaneous velocity) v(a) at time t = a to be the limit of these average velocities: This means that the velocity at time t = a is equal to the slope of the tangent line at P. 18 Example 3 Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground. (a) What is the velocity of the ball after 5 seconds? (b) How fast is the ball traveling when it hits the ground? Solution: We will need to find the velocity both when t = 5 and when the ball hits the ground, so it’s efficient to start by finding the velocity at a general time t = a. 19 Example 3 – Solution cont’d Using the equation of motion s = f(t) = 4.9t 2, we have 20 Example 3 – Solution cont’d (a) The velocity after 5 s is v(5) = (9.8)(5) = 49 m/s. (b) Since the observation deck is 450 m above the ground, the ball will hit the ground at the time t1 when s(t1) = 450, that is, 4.9t12 = 450 This gives t 12 = and t1 =  9.6 s The velocity of the ball as it hits the ground is therefore v(t1) = 9.8t1 = 9.8  94 m/s 21 Derivatives 22 Derivatives We have seen that the same type of limit arises in finding the slope of a tangent line (Equation 2) or the velocity of an object (Equation 3). In fact, limits of the form arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation. 23 Derivatives If we write x = a + h, then we have h = x – a and h approaches 0 if and only if x approaches a. Therefore an equivalent way of stating the definition of the derivative, as we saw in finding tangent lines, is 24 Example 4 Find the derivative of the function f(x) = x2 – 8x + 9 at the number a. Solution: From Definition 4 we have 25 Example 4 – Solution cont’d 26 Derivatives We defined the tangent line to the curve y = f(x) at the point P(a, f(a)) to be the line that passes through P and has slope m given by Equation 1 or 2. Since, by Definition 4, this is the same as the derivative f(a), we can now say the following. 27 Derivatives If we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve y = f(x) at the point (a, f(a)): y – f(a) = f(a)(x – a) 28 Rates of Change 29 Rates of Change Suppose y is a quantity that depends on another quantity x. Thus y is a function of x and we write y = f(x). If x changes from x1 to x2, then the change in x (also called the increment of x) is x = x2 – x1 and the corresponding change in y is y = f(x2) – f(x1) 30 Rates of Change The difference quotient is called the average rate of change of y with respect to x over the interval [x1, x2] and can be interpreted as the slope of the secant line PQ in Figure 8. Figure 8 31 Rates of Change By analogy with velocity, we consider the average rate of change over smaller and smaller intervals by letting x2 approach x1 and therefore letting Δx approach 0. The limit of these average rates of change is called the (instantaneous) rate of change of y with respect to x at x = x1, which is interpreted as the slope of the tangent to the curve y = f(x) at P(x1, f(x1)): We recognize this limit as being the derivative f(x1). 32 Rates of Change We know that one interpretation of the derivative f(a) is as the slope of the tangent line to the curve y = f(x) when x = a. We now have a second interpretation: The connection with the first interpretation is that if we sketch the curve y = f(x), then the instantaneous rate of change is the slope of the tangent to this curve at the point where x = a. 33 Rates of Change This means that when the derivative is large (and therefore the curve is steep, as at the point P in Figure 9), the y-values change rapidly. Figure 9 The y-values are changing rapidly at P and slowly at Q. 34 Rates of Change When the derivative is small, the curve is relatively flat (as at point Q) and the y-values change slowly. In particular, if s = f(t) is the position function of a particle that moves along a straight line, then f(a) is the rate of change of the displacement s with respect to the time t. In other words, f(a) is the velocity of the particle at time t = a. The speed of the particle is the absolute value of the velocity, that is, |f(a)|. 35 Example 6 A manufacturer produces bolts of a fabric with a fixed width. The cost of producing x yards of this fabric is C = f(x) dollars. (a) What is the meaning of the derivative f(x)? What are its units? (b) In practical terms, what does it mean to say that f(1000) = 9? (c) Which do you think is greater, f(50) or f(500)? What about f(5000)? 36 Example 6(a) – Solution The derivative f(x) is the instantaneous rate of change of C with respect to x; that is, f(x) means the rate of change of the production cost with respect to the number of yards produced. Because the units for f(x) are the same as the units for the difference quotient C/x. Since C is measured in dollars and x in yards, it follows that the units for f(x) are dollars per yard. 37 Example 6(b) – Solution cont’d The statement that f(1000) = 9 means that, after 1000 yards of fabric have been manufactured, the rate at which the production cost is increasing is $9/yard. (When x = 1000, C is increasing 9 times as fast as x.) Since x = 1 is small compared with x = 1000, we could use the approximation and say that the cost of manufacturing the 1000th yard (or the 1001st) is about $9. 38 Example 6(c) – Solution cont’d The rate at which the production cost is increasing (per yard) is probably lower when x = 500 than when x = 50 (the cost of making the 500th yard is less than the cost of the 50th yard) because of economies of scale. (The manufacturer makes more efficient use of the fixed costs of production.) So f(50) > f(500) 39 Example 6(c) – Solution cont’d But, as production expands, the resulting large-scale operation might become inefficient and there might be overtime costs. Thus it is possible that the rate of increase of costs will eventually start to rise. So it may happen that f(5000) > f(500) 40 2 Derivatives Copyright © Cengage Learning. All rights reserved.  Derivatives of Trigonometric 2.4 Functions Copyright © Cengage Learning. All rights reserved. Derivatives of Trigonometric Functions In particular, it is important to remember that when we talk about the function f defined for all real numbers x by f(x) = sin x it is understood that sin x means the sine of the angle whose radian measure is x. A similar convention holds for the other trigonometric functions cos, tan, csc, sec, and cot. All of the trigonometric functions are continuous at every number in their domains. 3 Derivatives of Trigonometric Functions If we sketch the graph of the function f(x) = sin x and use the interpretation of f(x) as the slope of the tangent to the sine curve in order to sketch the graph of f, then it looks as if the graph of f may be the same as the cosine curve. (See Figure 1). Figure 1 4 Derivatives of Trigonometric Functions Let’s try to confirm our guess that if f(x) = sin x, then f(x) = cos x. From the definition of a derivative, we have 5 Derivatives of Trigonometric Functions Two of these four limits are easy to evaluate. Since we regard x as a constant when computing a limit as h 0, we have and 6 Derivatives of Trigonometric Functions The limit of (sin h)/h is not so obvious. We made the guess, on the basis of numerical and graphical evidence, that 7 Derivatives of Trigonometric Functions We now use a geometric argument to prove Equation 2. Assume first that  lies between 0 and  /2. Figure 2(a) shows a sector of a circle with center O, central angle , and radius 1. BC is drawn perpendicular to OA. By the definition of radian measure, we have arc AB = . Also |BC| = |OB| sin  = sin . Figure 2(a) 8 Derivatives of Trigonometric Functions From the diagram we see that |BC| < |AB| < arc AB Therefore sin  <  so